This paper studies common unfoldings of various classes of polycubes, as well
as a new type of unfolding of polyominoes. Previously, Knuth and Miller found
a common unfolding of all tree-like tetracubes. By contrast, we show here
that all 23 tree-like pentacubes have no such common unfolding, although 22 of
them have a common unfolding. On the positive side, we show that there is an
unfolding common to all “non-spiraling” k-ominoes, a result
that extends to planar non-spiraling k-cubes.

Updates:

The phrase “There is a unique two-sided unfolding of all 22 non-planar pentacubes (Figure 8(12-27))” should be replaced with “There is a unique two-sided unfolding of all 16 non-planar pentacubes (Figure 8(12-27), which also folds into 6 planar pentacubes”.